Related papers: Bounding Vanishing at the Central Point of Cuspida…

The binary sum-of-digits function $\mathsf{s}$ returns the number of ones in the binary expansion of a nonnegative integer. Cusick's Hamming weight conjecture states that, for all integers $t\geq 0$, the set of nonnegative integers $n$ such…

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of so-called local polynomials and the action of finitely many Hecke operators thereon. Such polynomials are the…

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet…

We provide sufficient conditions under which the center-outward distribution and quantile functions introduced in Chernozhukov et al.~(2017) and Hallin~(2017) are homeomorphisms, thereby extending a recent result by Figalli \cite{Fi2}. Our…

The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset $S + S = \{x+y :…

Birch and Swinnerton-Dyer conjecture allows for sharp estimates on the rank of certain abelian varieties defined over $ \Q$. in the case of the jacobian of the modular curves, this problem is equivalent to the estimation of the order of…

Let F be a class of functions with the uniqueness property: if a function f in F vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a…

Realistic modeling of ecological population dynamics requires spatially explicit descriptions that can take into account spatial heterogeneity as well as long-distance dispersal. Here, we present Monte Carlo simulations and numerical…

We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg-Kohn functional for electronic or nucleonic densities and for the study…

In 2000 Iwaniec, Luo, and Sarnak proved for certain families of $L$-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density…

For plurisubharmonic functions $\varphi$ and $\psi$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $\varphi$ at the origin vanishes, we show that the mass of the origin with…

We prove Lindel\"of-on-average upper bounds on the cubic moment of central values of $L$-functions over certain families of $\operatorname{PGL}_2/\mathbb{Q}$ automorphic representations $\pi$ given by specifying the local representation…

For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp…

We consider two parametrized random digraph families, namely, proportional-edge and central similarity proximity catch digraphs (PCDs) and compare the performance of these two PCD families in testing spatial point patterns. These PCD…

We derive explicit upper bounds for the number of nondegenerate critical points of a $k$-component Gaussian mixture density in $\mathbb{R}^d$, and the number of modes when the modal set is finite, together with lower bounds. By normalizing…

We give a short proof of almost sure invertibility of unsymmetric random Kansa collocation matrices by a class of analytic RBF vanishing at infinity, for the Poisson equation with Dirichlet boundary conditions. Such a class includes popular…

This paper investigates asymptotic distribution of complex zeros of random polynomials $P_n(z):=\sum_{k=0}^{n}b(k)\xi_k z^k$, as $n\to\infty$, where $b$ is a regularly varying function at infinity with index $\alpha\in \mathbb{R}$ and…

Let f be an L^2-normalized Hecke--Maass cuspidal newform of level N and Laplace eigenvalue \lambda. It is shown that |f|_\infty <<_{\lambda, \epsilon} N^{-1/12 + \epsilon} for any \epsilon>0. The exponent is further improved in the case…

Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition…

This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level $\Gamma_0(4)$ and half-integral weights. Based on substantial calculations, the question is raised…